### Analytic Novikov for topologists (corrected version), by Jonathan Rosenberg

This is a corrected version of an earliar preprint in this archive.

We explain for topologists the dictionary'' for understanding the analytic proofs of the Novikov conjecture, and how they relate to the surgery-theoretic proofs. In particular, we try to explain the following points:

Why do the analytic proofs of the Novikov conjecture require the introduction of C*-algebras?

Why do the analytic proofs of the Novikov conjecture all use $K$-theory instead of $L$-theory? Aren't they computing the wrong thing? (In this connection we discuss in detail how the various (algebraic) $L$-theory spectra of a real or complex C*-algebra are related to its TOPOLOGICAL $K$-theory spectrum.)

How can one show that the index map $\mu$ or $\beta$ studied by operator theorists matches up with the assembly map in surgery theory?

Where does bounded surgery theory'' appear in the analytic proofs? Can one find a correspondence between the sorts of arguments used by analysts and the controlled surgery arguments used by topologists?

Jonathan Rosenberg <jmr@math.umd.edu>